## On the Subject of Partial Derivatives

You won't be getting partial credit...

This module displays a trinomial function in terms of three variables: **x**, **y**, and **z**. In each term of the trinomial, the variables are taken to integer powers between 0 and 5 (inclusive), and they may be multiplied by each other. Coefficients will appear on each term, and the terms will be added to each other. The module also contains a colored LED that may change color on each of the three stages.

The goal of this module is to evaluate a first, second, and third partial derivative of the function displayed based on the current color of the LED, and evaluate the partial derivative at points determined by the edgework of the bomb. Each of these partial derivatives is its own stage, and the current stage is indicated by the number of small white LEDs that are lit on the bottom of the module.

### How to take a partial derivative

The **derivative** of a function measures a function's output's sensitivity with respect to a change in its input. The derivative of a single-variable function evaluted at a point measures the function's instantaneous rate of change (slope) at that particular point.

While learning calculating derivatives takes up a sizeable portion of an introductory calculus class, only a single type of function will appear in this module: polynomials. For a monomial **f = Cx ^{n}**, the derivative

**D**is equal to

_{x}**Cnx**. The derivative of the sum of monomials is equal to the sum of the derivatives of the monomials. For example, if

^{n-1}**f = 6x**then

^{4}-x^{2}+9x-1,**D**. The value of the derivative of a function at a point can be found by plugging in the coordinates of the point into the derivative. For example, with the above function,

_{x}= 24x^{3}-2x+9**D**.

_{x}(1) = 24(1^{3})-2(1)+9 = 31The **partial derivative** of a multivariable function is the derivative of the function taken with respect to exactly one variable, treating all other variables as constants. For a partial derivative of **f** taken with respect to **x**, this is denoted **D _{x}f**. A second partial derivative can be taken with respect to the same or a different variable; for example, two second partial derivatives of

**f**are

**D**and

_{xx}f**D**. For example, for the function

_{xy}f**g = 2x**,

^{3}y^{2}-7xy^{4}**g**and

_{x}= 6x^{2}y^{2}-7y^{4}**g**. Evaluated at a point,

_{xy}= 12x^{2}y-28y^{3}**g**and

_{x}(1,2) = 6(1)^{2}(2)^{2}-7(2)^{4}= -88**g**. By Clairaut's Theorem,

_{xy}(1,2) = 12(1)^{2}(2)-28(2)^{3}= -200**D**.

_{xy}f = D_{yx}f